Download 1 Radial probability densities for electrons in an atom

Physics 365 - Many-Particle Quantum Theory - Prof. Oberacker
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Radial probability densities for electrons in an atom
In our lecture notes (chapter 3, p. 10) we derived a simple expression for the ground state density of
an N-electron atom if we neglect the electron-electron interaction. In this case the electron density
of an atom is simply the sum of the probabitity densities (= absolute value squared of the wave
functions) for all occupied single particle states.
The spatial part of the electronic wave functions has the structure
ψn,`,m` (r, θ, φ) = Rn,` (r) Y`,m` (θ, φ) .
In Niels Bohr’s semi-classical atomic model, the radii rn of the spherical electronic orbits increase
with the square of the principal quantum number n, i.e.
rn = n2 (
a0
),
Z
(n = 1, 2, 3, ...)
where a0 = 0.55 Å is the Bohr radius of the hydrogen atom, and Z denotes the atomic number (=
number of protons in the nucleus). In quantum mechanics, the radial probability densities
r2 [Rn,` (r)]2
are most closely related to these classical radii. The factor r2 in the last expression arises from the
volume element in spherical coordinates.
Below we plot the 1-D radial probability densities for the lowest quantum states with principle
quantum numbers n = 1, 2, 3.
atomic radial probability density (n=1)
L=0
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
r (units of Bohr radius / Z)
Figure 1: For n = 1, the probability density peaks at 1 Bohr radius / Z.
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atomic radial probability densities (n=2)
0.25
L=1
0.2
0.15
0.1
0.05
L=0
0
0
5
10
15
r (units of Bohr radius / Z)
Figure 2: For n = 2, the probability densities peak at 3-5 Bohr radii /Z (for the two angular
momentum substates).
atomic radial probability densities (n=3)
0.12
L=2
0.08
0.04
L=1
L=0
0
0
5
10
15
20
25
r (units of Bohr radius / Z)
Figure 3: For n = 3, the probability densities peak at 8-13 Bohr radii / Z (for the three angular
momentum substates).
From these plots we infer that the probability densities for n = 1, 2, 3 peak at different radial
positions which roughly agree with the semi-classical Bohr radii rn given above. This suggests that
the total density of an N-electron atom (which can be measured) might reveal the shell structure
of the occupied orbitals! This is indeed the case. We will look at atomic electron densities later in
Chapter 7 within the context of atomic Hartree-Fock calculations which take the electron-electron
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interaction into account “on average” (via the one-body mean field potential).
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Position probability densities in 3-D (‘atomic orbitals’)
We calculate now the position probability densities in 3-D space. In cylindrical coordinates (z, r, φ)
we have
ρn,`,m` (z, r, φ) = [ψ ∗ ψ]n,`,m` (z, r, φ) .
The probability densities are axially symmetric around the z-axis. In the contour plots below we
show the quantities
ρn,`,m` (z, r, φ = 0) .
prob_density_linear
3
0.31
0.286
2
0.263
0.239
1
0.215
z
0.191
0.167
0
0.143
0.119
1
0.0955
0.0717
2
0.0478
0.0239
3
6.57e 05
0
1
2
r
3
Figure 4: Probability density ψ ∗ ψ for the state |n = 1, ` = 0, m` = 0 > .
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prob_density_linear
11
10
0.00525
0.00242
0.00444
0.00222
0.00404
0.00202
0.00363
0.00182
0.00323
0.00162
0.00283
0
0.00262
0.00485
z
z
11
10
prob_density_linear
0.00141
0
0.00242
0.00121
0.00202
0.00101
0.00162
0.000808
0.00121
0.000606
0.000808
0.000404
0.000404
10
11
0.000202
10
11
0
0
r
1011
0
0
r
1011
Figure 5: Probability densities ψ ∗ ψ for the state |n = 2, ` = 1 > . Left side: for the magnetic
substate |m` = 0 > . Right side: for the magnetic substates |m` = ±1 > .
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prob_density_linear
35
0.00033
30
0.000304
0.000279
20
0.000254
0.000228
10
z
0.000203
0.000178
0
0.000152
0.000127
10
0.000101
7.61e 05
20
5.07e 05
2.54e 05
30
35
0
0
10
r
20
30 35
Figure 6: Probability density ψ ∗ ψ for the state |n = 4, ` = 2, m` = 0 > .
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